This study will develop statistical methods for analyzing categorical data, particularly in the comparison of different groups. Examples are the common odds ratios for a number of contingency tables and the standardized risk difference in the comparison of two groups stratified by a covariate. The optimality properties of the developed methodology will be investigated and established. The likelihood equations are to be derived. The newly developed, maximum likelihood and conditional maximum likelihood methodologies are to be examined. The study has been extended to do a comparison of a number of contingency tables in the form of estimation and hypothesis testing. This methodology is particularly useful in detecting the adequacy of matching in the case-control epidemiological studies. All these methodologies have wide applications to epidemiological studies, biomedical studies, genetics and other disciplines. Examples include testing the Hardy-Weinberg Law across strata in genetics and comparing the black-white differences in birth specific infant mortality curves. The study of the common odds ratio is extended to the case under cluster sampling, where the units within a cluster are not necessarily independent. Theoretical and simulation studies are conducted to validate the proposed methodology.